The successor monad as defined here is also interesting in that it stabilizes the finite von Neumann ordinals and monotone maps between them, and that η\eta and μ\mu are also monotone. Thus the successor monad restricts as a monad to the augmented simplex category. Furthermore, every monotone map of finite ordinals can be written as a composite of arrows of the form SkμlS^k \mu_l and SmηnS^m \eta_n. Indeed, the monoidal categoryΔa\Delta_a is generated by the monoid object0→ι01←μ020 \overset{\iota_0}\rightarrow 1 \overset{\mu_0}\leftarrow 2.

JCMcKeown: I want to say something like (S,η,μ)(S,\eta,\mu)generates the (skeletal augmented) simplex category; there is surely a right way to say that, but what is it?

Toby: I don't think that it's quite true that SS generates the simplex category, because you need an object to start applying it to. But I'd agree that (S,η,μ)(S,\eta,\mu) generates it starting from 00. I don't know any slick way to say that.